Introduction / Context:
This question asks you to find the original principal when you are given the total compound interest for a certain period and the rate of interest. Instead of giving the final amount, the problem provides only the interest earned over 2 years at compound interest. You need to reverse the formula to calculate the principal.
Given Data / Assumptions:
- Total compound interest over 2 years = Rs 3225.
- Rate of interest r = 15% per annum.
- Time period n = 2 years.
- Interest is compounded annually.
- Principal P is unknown and must be found.
Concept / Approach:
The amount A after 2 years at compound interest is A = P * (1 + r/100)^2. The compound interest CI is defined as CI = A - P. Since CI is given, we write P * [(1 + r/100)^2 - 1] = CI and solve this equation for P. This is a classic reverse compound interest calculation.
Step-by-Step Solution:
Step 1: Use the relation for compound interest over 2 years:
CI = P * [(1 + r/100)^2 - 1].
Step 2: Substitute r = 15 and CI = 3225:
3225 = P * [(1 + 15/100)^2 - 1].
Step 3: Compute the factor: 1 + 15/100 = 1.15.
Step 4: Square it: 1.15^2 = 1.3225.
Step 5: So [(1 + r/100)^2 - 1] = 1.3225 - 1 = 0.3225.
Step 6: Therefore 3225 = P * 0.3225, so P = 3225 / 0.3225 = 10000.
Verification / Alternative check:
Compute amount using P = 10000: A = 10000 * 1.3225 = 13225.
Compound interest = A - P = 13225 - 10000 = Rs 3225, which matches the given interest, confirming that the principal is correct.
Why Other Options Are Wrong:
A principal of 15000 or 20000 would produce interest values much larger than Rs 3225 at 15% for 2 years.
32250 clearly represents a large amount and not the required principal here.
Common Pitfalls:
An easy mistake is to treat Rs 3225 as the final amount rather than the interest. Another common error is to use the simple interest formula instead of compound interest, which ignores the interest on interest and leads to an incorrect principal.
Final Answer:
The principal that generates this compound interest is Rs 10000.
Discussion & Comments